Properties

Degree $2$
Conductor $100800$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 2·13-s − 6·17-s + 4·19-s − 6·29-s − 4·31-s + 2·37-s − 6·41-s + 8·43-s − 12·47-s + 49-s − 6·53-s − 12·59-s − 2·61-s + 8·67-s − 14·73-s − 16·79-s − 12·83-s − 6·89-s − 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.977·67-s − 1.63·73-s − 1.80·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{100800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8003944270\)
\(L(\frac12)\) \(\approx\) \(0.8003944270\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.64666528503757, −13.26662373348346, −12.81430237591720, −12.43834962211174, −11.60104353021035, −11.31134431984280, −10.93656974293193, −10.33449803725486, −9.591793173718489, −9.448870011173382, −8.741819920912246, −8.407299072109748, −7.609050128684146, −7.267195609963459, −6.626455010926036, −6.183750390004481, −5.596233750590144, −5.078074045251757, −4.315916283427464, −3.963529458355967, −3.130022679231671, −2.810257794946705, −1.800148028481237, −1.435167895458583, −0.2719925271679518, 0.2719925271679518, 1.435167895458583, 1.800148028481237, 2.810257794946705, 3.130022679231671, 3.963529458355967, 4.315916283427464, 5.078074045251757, 5.596233750590144, 6.183750390004481, 6.626455010926036, 7.267195609963459, 7.609050128684146, 8.407299072109748, 8.741819920912246, 9.448870011173382, 9.591793173718489, 10.33449803725486, 10.93656974293193, 11.31134431984280, 11.60104353021035, 12.43834962211174, 12.81430237591720, 13.26662373348346, 13.64666528503757

Graph of the $Z$-function along the critical line